May 10, 2017 - driven non-integrable system a MBL state may also arise. [41] and also ... n=1 Ïz n. (1). arXiv:1705.03662v1 [cond-mat.stat-mech] 10 May 2017 ...
Exact results in Floquet coin toss for driven integrable models
arXiv:1705.03662v1 [cond-mat.stat-mech] 10 May 2017
Utso Bhattacharya, Somnath Maity, Uddipan Banik and Amit Dutta Department of Physics, Indian Institute of Technology, Kanpur-208016, India We study an integrable Hamiltonian which is subjected to an imperfect periodic driving with the amplitude of driving (or kicking) randomly chosen from a binary distribution like a coin-toss problem. The randomness present in the driving protocol destabilises the periodic steady state, reached in the limit of perfectly periodic driving, leading to a monotonic rise of the stroboscopic residual energy with the number of periods. Remarkably, exploiting the completely uncorrelated nature of the randomness and the knowledge of the stroboscopic Floquet operator in the perfectly periodic situation, we find an exact analytical expression for the residual energy evaluated through a disorder operator. Furthermore, the stroboscopic average value of any local or global operator can be expressed in terms of the same. We further argue that this method is likely to work even for a non-integrable systems as long as there is a complete information of stroboscopic measurements.
The study of non-equilibrium dynamics of closed quantum systems is an exciting as well as a challenging area of recent research that has come to the fore, both from experimental [1–11] and theoretical perspectives [12–24]. One of the prominent areas in this regard happen to be studies of periodically driven closed quantum systems. (For review see, [25–31]) Although periodically driven systems have an illustrious history which dates back to the analysis of the famous Kapitza pendulum [32] and the kicked-rotor model [33], the recent interest in periodically driven systems are many fold: e.g., Floquet engineering of materials in their non-trivial phases such as the Floquet graphene [14, 15] and topological insulators [16] (see also [34]), dynamical generation of edge Majorana [19] and non-equilibrium phase transitions [35] like recently proposed time crystals [36, 37]. These studies have received a tremendous boost following experimental studies on light-induced non-equilibrium superconducting and topological systems [9, 10] and possibility of realising time crystals [38, 39]. The other relevant question deals with fundamental statistical aspects, namely the thermalisation of a closed quantum system under a periodic driving [40] and the possibility of the many-body localisation [41] (For a review on periodically driven quantum systems see [31]). Periodically driven closed quantum systems, from a statistical viewpoint, are being extensively studied in the context of defect and residual energy generation [40, 42], dynamical freezing [43], many-body energy localization [44] dynamical localisation[45, 46], dynamical fidelity [47], work-statistics [48, 49], and as well as in the context of entanglement entropy [50] and associated dynamical phase transitions [51]. For periodically driven closed integrable systems, it is usually believed that the system reaches a periodic steady state in the asymptotic limit of driving and hence stops absorbing energy [40, 53]: the resulting steady state can be viewed as a periodic Gaussian ensemble [52]. However, for a non-integrable model [54] or for an aperoidic driving of an integrable model, the system is expected to absorb energy indefinitely; also there
exists a possibility of a geometrical generalised Gaussian ensemble in some special situations [55]. However, for a driven non-integrable system a MBL state may also arise [41] and also a MBL state may get delocalised under a periodic driving [56]. In this paper, we consider a closed integrable quantum system undergoing an imperfect periodic dynamics. The imperfection or disorder manifests itself in the amplitude of the periodic drive which assumes binary values chosen from a binomial distribution resembling a series of biased coin toss events. The combination of a periodic driving with such a disordered amplitude results in the so-called “Floquet coin toss problem”. The aim of the paper is to provide an exact analytic framework to explore the statistical properties of such a non-equilibrium system observed at N -th stroboscopic interval determined by the inverse of the frequency (ω) of the monochromatic drive . Our study indeed confirms that the system, even though integrable, never reaches a periodic steady state and rather absorbs energy indefinitely due to the imperfectness present in the protocol. What is remarkable though is that using the uncorrelated nature of the amplitude for successive periods and stroboscopic information in the perfectly periodic situation, it is indeed possible to calculate any disorder-averaged (global or local) observable (for example, the residual energy) in terms of a given disorder operator that emerges through our calculation. The main goal of this work is to stroboscopically investigate the deviation of statistical observables under such an imperfect periodic drive from their behaviour under a perfectly periodic drive. To achieve this, we perform all our analysis in the basis of a perfectly periodic evolution operator (the Floquet evolution operator), and hence, complete stroboscopic information of the system over one drive cycle is of the essence here. We establish the above claim considering the one dimensional transverse Ising Model, described by the Hamiltonian, H=−
L X n=1
x τnx τn+1 −h
L X n=1
τnz .
(1)
2 where h is the transverse field and τni {i = x, y, z} are the Pauli spin matrices at nth site. This model is exactly solved via a Jordan-Wigner mapping of spin variables to spinless fermionic operators [27]; through the P transn−1 † formation relations, cn = exp πi j=1 aj aj an and Pn−1 c†n = a†n exp −πi j=1 a†j aj where an = 21 (τnx − iτny ). P Employing a Fourier transformation, cn = √1L k ck eink P and c†n = √1L k c†k e−ink , the Hamiltonian can be decoupled in Pto 2 × 2 problems for each Fourier mode such that H = k Hk with Hk = (h − cosk)σz + sinkσx where σ’s are again Pauli matrices. Using the anti-periodic boundary condition for the fermions with even number of L, possible values of the momentum modes are k = 2mπ L 1 1 L−1 with m = − L−1 2 , ..., − 2 , 2 , ..., 2 . It is well know that the system has quantum critical points at h = ±1. In our present work we study the effect of aperiodic temporal variation of the external transverse field h, considering two different types of driving protocols, namely, the delta kicks and the sinusoidal driving incorporating a binary disorder in the amplitude of driving. In short, we have h(t) = 1 + f (t) with f (t) =
N X
gn [αδ(t − nT )]
for the delta kick,
(2)
n=1
=
N X n=1
gn α sin
2πt T
for the sinusoidal drive, (3)
where α is the amplitude of driving and T is the time period. The random variable gn , takes the value either 1 with probability p or 0 with probability (1 − p) chosen from a Binomial distribution. Evidently gn = 0, corresponds to free evolution in the nth time period within the time interval (n − 1)T to nT while gn = 1 corresponds to the periodic perturbation in the form of a kick or sinusoidal driving. To illuminate the underlying Floquet theory, let us first consider the case of fully periodic situation (p = 1) choosing the initial state |ψk (0)i as the ground-state of the free Hamiltonian in our case; for each k mode, we then have Hk (t + T ) = Hk (t). Using the Floquet formalism, one can define a Floquet evolution operaRT tor Fk (T ) = T exp −i 0 Hk (t)dt , where T denotes the time ordering operator and the solutions of the Schr¨ odinger equation for a time periodic Hamiltonian (j) (j) (j) can be written as, |ψk (t)i = exp(−ik T )|φk (t)i. The (j) states |φk (t)i, the so called Floquet modes satisfying (j) (j) the condition |φk (t + T )i = |φk (t)i and the real quan(j) tities k are known as Floquet quasi-energies. In the case of the δ-function kick, Fk can be exactly defined, in the form of, Fk (T ) = exp(−iασz ) exp(−iHk0 T )
(4)
consisting of two pieces; the first one corresponds to the δ-kick at time t = T while the second part represents the free evolution generated by the time independent Hamiltonian Hk0 = (1 − cos k)σz + (sin k)σx from time 0 to T . However for the sinusoidal driving Fk (T ) can be numerically diagonalized to obtain the quasi-energies ± k and the corresponding Floquet modes |φ± i. Focussing on the mode k, after a time t = k P (j) −i(j) N T (j) k |φk (t)i, N T we have |ψk (N T )i = j=± rk e ± ± where rk = hφkP |ψk (0)i and hence the residual energy εres (N T ) = L1 k (ek (N T ) − ek (0)) with ek (N T ) = hψk (N T )|Hk0 |ψk (N T )i and ek (0) = hψk (0)|Hk0 |ψk (0)i. In the thermodynamic limit of L → ∞, we have X Z 1 0 α dk |rkα |2 hφα εres (N T ) = k |Hk |φk i 2π α=± X −β NT α∗ β i(α α 0 β ) k k + rk rk e hφk |Hk |φk i − ek (0) α,β=± α6=β
(5) In the limit N → ∞, due to the Riemann-Lebesgue lemma the rapidly oscillating off-diagonal terms in Eq. (5) drop off upon integration over all k modes leading to a steady state expression for εres [40]. Deviating from the completely periodic case and considering the situation 0 < p < 1, so that we have a probability (1 − p) of missing a kick (or a cycle of sinusoidal drive is absent) in every complete period, one can now write the corresponding evolved state after N complete periods as |ψk (N T )i = Uk (gN )Uk (gN −1 ).......Uk (g2 )Uk (g1 )|ψk (0)i (6) with the generic evolution operator given by, ( Fk (T ), if gn = 1. Uk (gn ) = (7) Uk0 (T ), if gn = 0. where Fk (T ) is the usual Floquet operator and Uk0 (T ) = exp(−iHk0 T ) is the time evolution operator for the free Hamiltonian Hk0 . One then readily finds: ek (N T ) = hψk (0)|U†k (g1 )U†k (g2 ).......U†k (gN −1 )U†k (gN ) × Hk0 Uk (gN )Uk (gN −1 )........Uk (g2 )Uk (g1 )|ψk (0)i (8) The numerical calculation in Eq. (8) thus involves the multiplication of (2N + 1) matrices corresponding to N complete periods for a given disorder configuration followed by configuration averaging over the disorder. Let us refer to the Fig. 1, where numerically obtained residual energy (RE) is plotted as a function of number of complete periods N choosing a high frequency limit
3 0.0006
Exact Results Numerical Results
0.6
●
⋄⋄⋄⋄ ⋄⋄⋄⋄++++ ⋄ ⋄ ⋄ ⋄⋄ ++++ + p=0.1 p=0.1 ⋄⋄⋄ + p=0.3 ⋄⋄⋄++++ + + ⋄ p=0.3 p=0.5 ⋄ ⋄ ++ ⋄ ^^^^ ⋄ + + p=0.5 ^^^ p=0.7 ⋄⋄++++ ^^^ ^ ⋄ ^ p=0.9 ^ p=0.7 ⋄ ++ ^ ^^ + p=1 ^^^ ⋄⋄ ++ ^^^ * p=0.9 ^ + ⋄⋄ ^ ⋄++ ^ ^ ^ ^ p=1 ⋄ + ^ ***** ^ + ⋄ ******* ^ + ^ ⋄ ****** +^ ^ ^ * * * * * ⋄ * +^ + ****** ^^ ^ * ^⋄ + *⋄ * * * * * * ^ + ■■ * * ■■■ ■■■ ⋄ ■■■ + ⋄ ■■■ ■ ■ ■■ ○
p=0
p=0
■
p=0.01
p=0.01
●
0.4
●
●
● ●
●
●
0.2
●
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ●●●●●●● ●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
●
0.0
■■ ■■■ ■■■ ■■■ ■ ■ ■■ ■■ ■■ ■■ ■○ ○ ■○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ + ^
⋄ *
200
400
600
800
Numerical Results
○
p=0
p=0
■
p=0.01
p=0.01
p=0.1
p=0.1
⋄
p=0.3
p=0.3
+ ^
p=0.5
p=0.7
p=0.7
p=0.9
*
p=0.9
p=0.5
p=1
+ ^
+ ++ + + ^^ ++ ^ ^ ^ + ^ ++ ^ ^ ⋄⋄ ^^ ++ ⋄⋄ ^ ⋄ + ^ ⋄ ++ ^^ ⋄⋄ + ^^ + ⋄⋄ ^ ⋄ + ^ ⋄
⋄ ** ⋄⋄ **** ⋄⋄ * * * * * * * * ⋄ ^ ⋄ ** ^+ ^+ **** *⋄ ^+ *⋄ *⋄ ^+ *⋄ * * ^ 0.0002 ⋄ * + * * * ⋄⋄ ^ *^ + *+ ⋄ * * * * * *^ *^ *^ + + ⋄⋄⋄ ^ + ^ ⋄ ^ ^ ^ +++ ⋄⋄ 0.0001 + ⋄⋄ + ⋄ + ⋄ ⋄ ⋄⋄ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 0.0000 ⋄ ■ ■ ○ ■ ○ ○ + ^ * ○■ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 0.0003
^ ^+ ^+ +
p=1
●
+ + ^^
●
●
●
●
●
●
● ●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
0
0.0004
●
●
Exact Results
0.0005
Residual Energy (RE)
Residual Energy (RE)
0.8
0
1000
No. of stroboscopic intervals (N)
200
400
600
800
1000
No. of stroboscopic intervals (N)
(a)
(b)
FIG. 1: (Color online) (a) The residual energy plotted as a function of stroboscopic intervals (N ) for a randomly kicked transverse Ising chain for α = π/16. The solid lines correspond to numerically obtained results for different values of p, while different symbols represent corresponding exact analytical results. (b) The same for the random sinusoidal driving for α = 1. In both the cases, driving frequency ω is chosen to be 100 and the number of configurations used in numerics Nc = 1000. For the fully periodic situation (p = 1), the system synchronises with the external driving and stops absorbing energy. On the contrary for any non-zero value of p 6= 1, the periodic steady state gets destabilised and the system absorbs heat indefinitely. Furthermore, the value of p for which the rate of rise of RE with N will be asymptotically maximum [around p = 0.3 in (a) and p = 0.5 in (b)] depends on both α and ω of the imperfect drive.
(ω = 2π/T > 4) in which the fully periodic situation leads to a periodic steady state both for the δ-kicks and sinusoidal variation and the system synchronises with the external driving. Observing the monotonic rise of RE with N for a given p 6= 0, 1, we conclude that for any non-zero value of p, the periodic steady state gets destabilised and system keeps absorbing energy indefinitely. We further note, that for a given N , the behavior of the RE cannot monotonically rise with p since the RE gets constrained by the fully periodic situation around p = 1 and the no rise situation around p = 0. The value of p for which the rate of rise of RE with N will be asymptotically maximum depends on both α and ω of the imperfect
" N X Y (0) (N ) (N ) (0) hek (N T )i = / hψk (0)|jk ihjk |Hk0 |ik ihik |ψk (0)i m=1
drive.
Having provided the numerical results, we shall proceed to set up the corresponding analytical framework within the space spanned by the complete set of Floquet basis {|jk± i} states. Introducing 2(N + 1) identity operators in terms of the Floquet basis states P (m) (m) (m) ˆ can take j (m) |jk ihjk | = 1 in Eq. (8), where jk two possible values corresponding to two quasi-states of the 2 × 2 Floquet Hamiltonian Fk (T ) for each mode k and performing the average over disordered configurations and finally upon reorganisation, we find,
!# X
(m−1) (m) (m) (m−1) P (gm )hjk |U†k (gm )|jk ihik |Uk (gm )|ik i
gm =1,0
(9)
P P where / ≡ j (0) ,j (1) ,....,j (N ) . The uncorrelated nature i
(0)
(1)
,i
(N )
,....,i
of gm s enables us to perform the configuration average by separately averaging over for each gm . Recalling ± Eq. (7), and the fact that Fk (T )|jk± i = exp(−i± k T )|jk i
and P (gm ) is the probability of being perfectly driven [P (gm = 1) = p] and free evolution [P (gm = 0) = (1−p)], respectively, leads us to:
4
X (0) (N ) (N ) (0) hek (N T )i = / hψk (0)|jk ihjk |Hk0 |ik ihik |ψk (0)i " N # j i Y iT km−1 −km−1 (m−1) (m) 0† (m) 0 (m−1) |Uk |jk ihik |Uk |ik i × pe δj (m−1) ,j (m) δi(m) ,i(m−1) + (1 − p)hjk k
m=1
k
k
k
=
X
(0)
(N )
hψk (0)|jk ihjk
X
(0)
(N )
hψk (0)|jk ihjk
,j (m) ,i(m) ,i(m−1)
,j (m) ,i(m) ,i(m−1)
≡
pe
j i iT km−1 −km−1
(10)
j (0) ,j (N ) ,i(N ) ,i(0)
where the matrix element of (4 × 4) matrix Dk is given
(m−1)
(m−1)
Dkj
i h (0) (0) |Hk |i(N ) ihik |ψk (0)i DN k
j (0) ,j (N ) i(0) ,i(N )
Dkj
N Y
X
j (1) ,j (2) .....,j (N −1) m=1 i(1) ,i(2) ,.....,i(N −1)
j (0) ,j (N ) i(0) ,i(N )
=
(N ) (0) |Hk0 |ik ihik |ψk (0)i
by,
(m−1)
δj m−1 ,j m δim ,im−1 + (1 − p)hjk k
k
It should be noted that in Eq. (10), the N in DN k is not a label but the matrix Dk that we have defined in Eq. (11), raised to the power N . The above exercise naturally leads to the emergence of 4 × 4 disorder matrix D for a imperfectly driven 2 × 2 system. Given the amplitude, frequency, dimensionality, the form of Fk (T ) in every stroboscopic intervals and the knowledge of disorder encoded in the probability p of driving, every element of D-matrix can be exactly calculated. Thus, we have constructed a complete analytic framework to deal with any “Floquet coin toss” problems in general. The RE as obtained from the exact form given in Eq. 10 confirms the numerically obtained values presented in Fig. 1. The numerical simulations addressing similar problems involve multiplication of N unitary matrices, followed by averaging over a large number (Nc ) of disorder configurations; as N and Nc → ∞, the numerical method not only requires huge computational time but also becomes susceptible to inaccuracies creeping in through numerous matrix multiplications. On the contrary, the configuration averaging over disorder is automatically taken care of in the analytical framework which is evidently a nonperturbative method and hence valid for any amplitude and any range of driving frequency. This needs to be further emphasised that the analytical structure we frame is not only restricted to integrable model reducible to 2 × 2 problems for each k: rather, it can be generalised to non-integrable situations given uncorrelated binary distribution of disorder and complete
k
k
(m)
|U0† k |jk
(m)
(m−1)
ihik |U0k |ik
i
(11)
stroboscopic information about the Floquet operator at stroboscopic times. The coin toss situation proposed here can be experimentally verified as in the kicked rotor experiment designed by Sarkar et al. [57], where the decoherence has been induced by suppressing kicks entirely at certain time instants dictated by the value of waiting time between successive kicks drawn from Levy distribution. Given a rare possibility of analytical approach to explore a temporally disordered situation and given the wide scope of the validity of our results, we believe that our approach is going to provide a new avenue to a plethora of similar studies. AD acknowledges SERB, DST, New Delhi for financial support. We acknowledge Diptiman Sen for critical comments.
[1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). [2] M. Lewenstein, A. Sanpera, and V. Ahufinger, (Oxford University Press, Oxford (2012)). [3] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif and T. Esslinger, Nature 515, 237 (2014). [4] M. Greiner , O. Mandel, T. W. Hansch and I. Bloch, Nature 419, 51 (2002). [5] T. Kinoshita, T. Wenger and D. S. Weiss, Nature 440, 900 (2006). [6] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B.
5
[7] [8]
[9]
[10]
[11]
[12] [13] [14] [15] [16] [17] [18]
[19] [20] [21] [22] [23] [24] [25] [26] [27]
[28] [29] [30]
[31]
Rauer, M. Schreitl, I. Mazets1, D. Adu Smith, E. Demler, and J. Schmiedmayer, Science 337, 1318 (2012). S. Trotzky, Y-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwck, J. Eisert and I. Bloch, Nature 8, 325 (2012). M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauss, T. Fukuhara, C. Gross, I. Bloch, C. Kollath and S. Kuhr, Nature 481, 484 (2012). D. Fausti, R. I. Tobey, , N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi,4, A. Cavalleri, Science 331, 189 (2011). M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D.Podolsky, F. Dreisow, S. Nolte, M. Segev and A. Szameit, Nature 496 196 (2013). M. Schreiber, S. S. Hodgman, P. Bordia, Henrik P. Lschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, I. Bloch, Science 349, 842 (2015). P. Calabrese, and J. Cardy, Phys. Rev. Lett. 96, 136801 (2006); J. Stat. Mech, P06008 (2007). M. Rigol, V. Dunjko and M. Olshanii, Nature 452, 854 (2008). T Oka, H Aoki, Phys. Rev. B 79 081406 (2009). T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010). N. H. Lindner, G. Refael and V. Galitski, Nat. Phys. 7, 490-495, (2011). A. Bermudez, D. Patane, L. Amico, M. A. MartinDelgado, Phys. Rev. Lett. 102, 135702, (2009). A. A. Patel, S. Sharma, A. Dutta, Eur. Phys. Jour. B 86, 367 (2013); A. Rajak and A. Dutta, Phys. Rev. E 89, 042125, 2014. P. D. Sacramento, Phys. Rev. E 90 032138, (2014); M. D. Caio, N. R. Cooper and M. J. Bhaseen, Phys. Rev. Lett. 115, 236403 (2015). M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, Phys. Rev. B 88, 155133 (2013). A Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010). R. Nandkishore, D. A. Huse, Annual Review of Condensed Matter Physics, 6, 15-38 (2015). M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett., 110, 135704 (2013). J. C. Budich and M. Heyl, Phys. Rev. B 93, 085416 (2016). S. Sharma, U. Divakaran, A. Polkovnikov and A. Dutta, Phys. Rev. B 93, 144306 (2016). J. Dziarmaga, Advances in Physics 59, 1063 (2010). A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011). A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. Rosenbaum and D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, 2015). J. Eisert, M. Friesdorf and C. Gogolin, Nat. Phys. 11, 124 (2015). L. D’Alessio, Y. Kafri, A. Polkovnikov, M. Rigol, Adv. Phys. 65, 239 (2016). J. Stat. Mech.: Theo. and Expt, special issueQuantum Integrability in Out of Equilibrium Systems edited by P. Calabrese., F. H. L. Essler and G. Mussardo, 2016 (2016). M. Bukov, L. D’Alessio and A. Polkovnikov, Adv. Phys.
64 , No. 2, 139-226 (2016). [32] P. L. Kapitza. Soviet Phys. JETP, 21, 588, (1951). [33] G. Casati, B. V. Chirikov, F. M. Izrailev, and J. Ford, Lecture Notes in Physics, Springer N.Y., 93, 334 (1979). [34] J. Cayssol, B. Dora, F. Simon and R. Moessner, Phys. Status Solidi RRL, 7, 101-108 (2013). [35] V. M. Bastidas, C. Emary, B. Regler, T. Brandes, Phys. Rev. Lett. 108, 043003 (2012); V. M. Bastidas, C. Emary, G. Schaller, T. Brandes, Phys. Rev. A 86, 063627 (2012); Sayak Dasgupta, Utso Bhattacharya, Amit Dutta, Phys. Rev. E 91, 052129 (2015);Lorenzo Privitera, Giuseppe E. Santoro, Phys. Rev. B 93, 241406(R) (2016). [36] D. V. Else, B. Bauer, and C. Nayak, Phys. Rev. Lett. 117, 090402 (2016). [37] V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, S. L. Phys. Rev. Lett. 116, 250401 (2016). [38] J. Zhang, et al., Nature 543, 217( 2017). [39] S. Choi, etal., Nature 543, 221 (2017). [40] A. Russomanno, A. Silva, and G. E. Santoro, Phys. Rev. Lett. 109, 257201 (2012). [41] Pedro Ponte, Z. Papi, Franois Huveneers, Dmitry A. Abanin, Phys. Rev. Lett. 114, 140401 (2015); Pedro Ponte, Anushya Chandran, Z. Papi, Dmitry A. Abanin, Annals of Physics 353, 196 (2015) [42] V. Mukherjee, A. Dutta, and D. Sen, Phys. Rev. B 77, 214427 (2008); V. Mukherjee and A. Dutta, J. Stat. Mech. (2009) P05005. [43] A. Das, Phys. Rev. B 82, 172402 (2010). [44] L. DAlessio and A. Polkovnikov, Ann. Phys. (N.Y.) 333, 19 (2013). [45] T. Nag, S. Roy, A. Dutta, and D. Sen, Phys. Rev. B 89, 165425 (2014); [46] A. Agarwala, U. Bhattacharya, A. Dutta, D. Sen, Phys. Rev. B 93,174301 (2016); A. Agarwala, D. Sen,Phys. Rev. B 95, 014305 (2017). [47] S. Sharma, A. Russomanno, G. E. Santoro, and A. Dutta, EPL 106, 67003 (2014). [48] Anirban Dutta, Arnab Das, K. Sengupta, Phys. Rev. E 92, 012104 (2015). [49] A. Russomanno, S. Sharma, A. Dutta and G. E. Santoro, J. Stat. Mech. (2015) P08030. [50] Angelo Russomanno, Giuseppe E. Santoro, Rosario Fazio, J. Stat. Mech. (2016) 073101. [51] Arnab Sen, Sourav Nandy, K. Sengupta, Phys. Rev. B 94, 214301 (2016). [52] A. Lazarides, A. Das, and R. Moessner, Phys. Rev. Lett. 112, 150401 (2014). [53] V. Gritsevand A. Polkovnikov, arXiv:1701.05276 (2017). [54] Luca D’Alessio, Marcos Rigol, Phys. Rev. X 4, 041048 (2014). [55] Sourav Nandy, Arnab Sen, Diptiman Sen, arXiv: 1701.07596 (2017). [56] Achilleas Lazarides, Arnab Das, Roderich Moessner, Phys. Rev. Lett. 115, 030402 (2015); Asmi Haldar, Arnab Das, arXiv:1702.03455 (2017). [57] S. Sarkar, S. Paul, C. Vishwakarma, S. Kumar, G. Verma, M. Sainath, Umakant D. Rapol, M. S. Santhanam, hys. Rev. Lett. 118, 174101 (2017).
